5.4 Properties of Logarithms

3/1/2013

Properties of Logarithms

Let m and n be positive numbers and b ≠ 1,Product Property

Quotient Property

Power Property

nmmn bbb logloglog

nmn

mbbb logloglog

mnm bn

b loglog

Example 1 Expand a Logarithmic Expression

Expand the expression. Assume all variables are positive.

a. log4 5x 2 log7 y3x

b.

SOLUTION

a. log4 5x 2 = log4 5 log4 x 2+ Product property

= log4 5 2 log4 x+ Power property

log7 y3x

b. = log7 3x log7 y– Quotient property

= log7 3 log7 x log7 y– Product property+

CheckpointExpand and Condense Logarithmic Expressions

Expand the expression.

5. log2 5x

6. log 2x 3 ANSWER log 2 + 3 log x

ANSWER log2 5 + log2 x

7. log3 7

5xANSWER log3 5 –+ log3 x log3 7

8. log6 y4x 2

ANSWER log6 4 –+ 2 log6 x log6 y

Example 2 Solve for x

Find the value of x.

a. log5 125 = x

Logx 64 = 3b.

Rewrite in Exp. Form: (5 raised to what power equals 125?)

x = 3

Rewrite in Exp. Form: (what do you raised to 3rd power to get 64?)

x = 4

Example 2 Solve for x

Find the value of x.

c. log3 x = 4

ln x = 4d.

Rewrite in Exp. Form: (3 to the 4th power equals what?)

x = 81

Rewrite in Log. Form: Rewrite in Exp. Form:

x =

ln x = -1e.

Rewrite in Log. Form: Rewrite in Exp. Form:

x =

Homework:WS 5.4 #1-7odd, 8-17 all,19-29 odd